52 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    A general approach to transversal versions of Dirac-type theorems

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    Given a collection of hypergraphs H=(H1,,Hm)\textbf{H}=(H_1,\ldots,H_m) with the same vertex set, an mm-edge graph Fi[m]HiF\subset \cup_{i\in [m]}H_i is a transversal if there is a bijection ϕ:E(F)[m]\phi:E(F)\to [m] such that eE(Hϕ(e))e\in E(H_{\phi(e)}) for each eE(F)e\in E(F). How large does the minimum degree of each HiH_i need to be so that H\textbf{H} necessarily contains a copy of FF that is a transversal? Each HiH_i in the collection could be the same hypergraph, hence the minimum degree of each HiH_i needs to be large enough to ensure that FHiF\subseteq H_i. Since its general introduction by Joos and Kim [Bull. Lond. Math. Soc., 2020, 52(3):498-504], a growing body of work has shown that in many cases this lower bound is tight. In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac-type results for (powers of) Hamilton cycles. For example, we derive that any collection of rnrn graphs on an nn-vertex set, each with minimum degree at least (r/(r+1)+o(1))n(r/(r+1)+o(1))n, contains a transversal copy of the rr-th power of a Hamilton cycle. This can be viewed as a rainbow version of the P\'osa-Seymour conjecture.Comment: 21 pages, 4 figures; final version as accepted for publication in the Bulletin of the London Mathematical Societ

    The hitting time of clique factors

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    In a recent paper, Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let r3r \ge 3 and let nn be divisible by rr. Then, in the random rr-uniform hypergraph process on nn vertices, as soon as the last isolated vertex disappears, a perfect matching emerges. In the present work, we transfer this hitting time result to the setting of clique factors in the random graph process: At the time that the last vertex joins a copy of the complete graph KrK_r, the random graph process contains a KrK_r-factor. Our proof draws on a novel sequence of couplings, extending techniques of Riordan and the first author. An analogous result is proved for clique factors in the ss-uniform hypergraph process (s3s \ge 3)

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Paths and cycles in graphs and hypergraphs

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    In this thesis we present new results in graph and hypergraph theory all of which feature paths or cycles. A kk-uniform tight cycle Cn(k)C^{(k)}_n is a kk-uniform hypergraph on nn vertices with a cyclic ordering of its vertices such that the edges are all kk-sets of consecutive vertices in the ordering. We consider a generalisation of Lehel's Conjecture, which states that every 2-edge-coloured complete graph can be partitioned into two cycles of distinct colour, to kk-uniform hypergraphs and prove results in the 4- and 5-uniform case. For a kk-uniform hypergraph~HH, the Ramsey number r(H){r(H)} is the smallest integer NN such that any 2-edge-colouring of the complete kk-uniform hypergraph on NN vertices contains a monochromatic copy of HH. We determine the Ramsey number for 4-uniform tight cycles asymptotically in the case where the length of the cycle is divisible by 4, by showing that r(Cn(4))r(C^{(4)}_n) = (5+oo(1))nn. We prove a resilience result for tight Hamiltonicity in random hypergraphs. More precisely, we show that for any γ\gamma >0 and kk \geq 3 asymptotically almost surely, every subgraph of the binomial random kk-uniform hypergraph G(k)(n,nγ1)G^{(k)}(n, n^{\gamma -1}) in which all (k1)(k-1)-sets are contained in at least (12+2γ)pn(\frac{1}{2}+2\gamma)pn edges has a tight Hamilton cycle. A random graph model on a host graph HH is said to be 1-independent if for every pair of vertex-disjoint subsets A,BA,B of E(H)E(H), the state of edges (absent or present) in AA is independent of the state of edges in BB. We show that pp = 4 - 23\sqrt{3} is the critical probability such that every 1-independent graph model on Z2×Kn\mathbb{Z}^2 \times K_n where each edge is present with probability at least pp contains an infinite path

    A general approach to transversal versions of Dirac-type theorems

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    Given a collection of hypergraphs =(1,...,) with the same vertex set, an -edge graph ⊂∪∈[] is atransversal if there is a bijection ∶()→[] such that ∈(()) for each ∈(). How large does the minimum degree of each need to be so that necessarily contains a copy of that is a transversal? Each in the collection could be the same hypergraph,hence the minimum degree of each needs to be large enough to ensure that ⊆. Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020)498–504), a growing body of work has shown that inmany cases this lower bound is tight. In this paper, wegive a unified approach to this problem by providinga widely applicable sufficient condition for this lowerbound to be asymptotically tight. This is general enoughto recover many previous results in the area and obtainnovel transversal variants of several classical Dirac-typeresults for (powers of) Hamilton cycles. For example, wederive that any collection of graphs on an -vertex set, each with minimum degree at least (∕( + 1) +(1)), contains a transversal copy of the th power of a Hamilton cycle. This can be viewed as a rainbow versionof the Pósa–Seymour conjecture

    A general approach to transversal versions of Dirac-type theorems

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    Given a collection of hypergraphs (Formula presented.) with the same vertex set, an (Formula presented.) -edge graph (Formula presented.) is a transversal if there is a bijection (Formula presented.) such that (Formula presented.) for each (Formula presented.). How large does the minimum degree of each (Formula presented.) need to be so that (Formula presented.) necessarily contains a copy of (Formula presented.) that is a transversal? Each (Formula presented.) in the collection could be the same hypergraph, hence the minimum degree of each (Formula presented.) needs to be large enough to ensure that (Formula presented.). Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020) 498–504), a growing body of work has shown that in many cases this lower bound is tight. In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac-type results for (powers of) Hamilton cycles. For example, we derive that any collection of (Formula presented.) graphs on an (Formula presented.) -vertex set, each with minimum degree at least (Formula presented.), contains a transversal copy of the (Formula presented.) th power of a Hamilton cycle. This can be viewed as a rainbow version of the Pósa–Seymour conjecture

    Resilience for Loose Hamilton Cycles

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    We study the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum dd-degree threshold for loose Hamiltonicity relative to the random kk-uniform hypergraph Hk(n,p)H_k(n,p) coincides with its dense analogue whenever pn(k1)/2+o(1)p \geq n^{- (k-1)/2+o(1)}. The value of pp is approximately tight for d>(k+1)/2d>(k+1)/2. This is particularly interesting because the dense threshold itself is not known beyond the cases when dk2d \geq k-2.Comment: 33 pages, 3 figure

    Perfect matchings in random sparsifications of Dirac hypergraphs

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    For all integers nk>d1n \geq k > d \geq 1, let md(k,n)m_{d}(k,n) be the minimum integer D0D \geq 0 such that every kk-uniform nn-vertex hypergraph H\mathcal H with minimum dd-degree δd(H)\delta_{d}(\mathcal H) at least DD has an optimal matching. For every fixed integer k3k \geq 3, we show that for nkNn \in k \mathbb{N} and p=Ω(nk+1logn)p = \Omega(n^{-k+1} \log n), if H\mathcal H is an nn-vertex kk-uniform hypergraph with δk1(H)mk1(k,n)\delta_{k-1}(\mathcal H) \geq m_{k-1}(k,n), then a.a.s.\ its pp-random subhypergraph Hp\mathcal H_p contains a perfect matching (mk1(k,n)m_{k-1}(k,n) was determined by R\"{o}dl, Ruci\'nski, and Szemer\'edi for all large nkNn \in k\mathbb N). Moreover, for every fixed integer d<kd < k and γ>0\gamma > 0, we show that the same conclusion holds if H\mathcal H is an nn-vertex kk-uniform hypergraph with δd(H)md(k,n)+γ(ndkd)\delta_d(\mathcal H) \geq m_{d}(k,n) + \gamma\binom{n - d}{k - d}. Both of these results strengthen Johansson, Kahn, and Vu's seminal solution to Shamir's problem and can be viewed as "robust" versions of hypergraph Dirac-type results. In addition, we also show that in both cases above, H\mathcal H has at least exp((11/k)nlognΘ(n))\exp((1-1/k)n \log n - \Theta (n)) many perfect matchings, which is best possible up to a exp(Θ(n))\exp(\Theta(n)) factor.Comment: 25 pages + 2 page appendix; Theorem 1.5 was proved in independent work of Pham, Sah, Sawhney, and Simkin (arxiv:2210.03064
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